It's easiest to address these three sentences individually.
Robert and his family will move to the country and live there for a short period of time.
This is dependent on context. As it is, I would perceive 'the country' as equivalent to 'the countryside' as defined on Wiktionary:
(usually preceded by “the”) A rural area, as opposed to a town or city; the countryside.
As such, this would not be interchangeable with nation. However, if you had a context like.
Italy is so beautiful at this time of year. Robert and his family will move to the country and live there for a short period of time.
Then you might be able to get away with changing the sense of 'the country' to pertain to Italy, but I'd avoid this; it introduces ambiguity.
Will Rob return to his home country?
In this sense, yes, you could change 'country' for 'nation'. Though in the UK at least, 'home country' would be a more standard way of expressing it. 'home nation' would still be understood, but there's also a separate meaning of 'the Home Nations'.
He will live in a remote country where has fewer friends.
Yes, 'nation' could be used instead.
Generally speaking, they would be considered synonyms outside of the aforementioned sense (and when referring to 'country music').
However, there are also some instances where the terms can be contrasted and 'nation' refers to a people, while 'country' refers to their territory. For example, Wiktionary presents the sentence:
The Roma are a nation without a country.
And if referring to a specific country, the word can suggest legal recognition as a sovereign state, as discussed on Wikipedia:
The term "country" is generally used to refer to sovereign states.
Whereas 'nation' is not so constrained.